Theorem sbcieg 3756

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.) Avoid ax-10 2137, ax-12 2171. (Revised by Gino Giotto, 12-Oct-2024.)

Hypothesis

Ref	Expression
sbcieg.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
sbcieg	⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem sbcieg

Step	Hyp	Ref	Expression
1		df-sbc 3717	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
2		sbcieg.1	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3	2	elabg 3607	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
4	1, 3	bitrid 282	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2106  {cab 2715  [wsbc 3716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-sbc 3717

This theorem is referenced by:  sbcie  3759  2nreu  4375  reuprg0  4638  rabsnif  4659  ralrnmptw  6970  ralrnmpt  6972  fpwwe2lem3  10389  nn1suc  11995  opfi1uzind  14215  mndind  18466  fgcl  23029  cfinfil  23044  csdfil  23045  supfil  23046  fin1aufil  23083  ifeqeqx  30885  nn0min  31134  bnj1452  33032  cdlemk35s  38951  cdlemk39s  38953  cdlemk42  38955  2nn0ind  40767  zindbi  40768  prproropreud  44961